Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9,11,23)(6,17)(10,22,26,33)(12,24,13,18)(15,30)(19,21)(25,31,27,34)(28,32);(3,4,7,12)(5,10,16,27)(6,18,17,23)(8,13,20,11)(9,21,24,19)(14,25,29,26)(15,31,30,33)(22,32,34,28);(3,5,7,16)(4,10,12,27)(6,15,17,30)(8,14,20,29)(9,22,24,34)(11,26,13,25)(18,31,23,33)(19,28,21,32);(5,14)(10,25)(15,28)(16,29)(22,31)(26,27)(30,32)(33,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250629);